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The complex inverse trigonometric and hyperbolicfunctions
Howard E. Haber
Santa Cruz Institute for Particle Physics
University of California, Santa Cruz, CA 95064, USA
September 10, 2012
Abstract
In these notes, we examine the inverse trigonometric and hyperbolic func-tions, where the arguments of these functions can be complex numbers. Themulti-valued functions are defined in terms of the complex logarithm. We alsocarefully define the corresponding single-valued principal values of the inversetrigonometric and hyperbolic functions following the conventions employed bythe computer algebra software system, Mathematica 8.
1 Introduction
The inverse trigonometric and hyperbolic functions evaluated in the complex planeare multi-valued functions (see e.g. ref. 1). In many applications, it is convenientto define the corresponding single-valued functions, called the principal values ofthe inverse trigonometric and hyperbolic functions, according to some convention.Different conventions appear in various reference books. In these notes, we shall followthe conventions employed by the computer algebra software system, Mathematica 8,which are outlined in section 2.2.5 of ref. 2.
The principal value of a multi-valued complex function f(z) of the complex vari-able z, which we denote by F (z), is continuous in all regions of the complex plane,except on a specific line (or lines) called branch cuts. The function F (z) has a dis-continuity when z crosses a branch cut. Branch cuts end at a branch point, whichis unambiguous for each function F (z). But the choice of branch cuts is a matterof convention. Thus, if mathematics software is employed to evaluate the functionF (z), you need to know the conventions of the software for the location of the branchcuts. The mathematical software needs to precisely define the principal value of f(z)in order that it can produce a unique answer when the user types in F (z) for a par-ticular complex number z. There are often different possible candidates for F (z) thatdiffer only in the values assigned to them when z lies on the branch cut(s). Thesenotes provide a careful discussion of these issues as they apply to the complex inversetrigonometric and hyperbolic functions.
The simplest example of a multivalued function is the argument of a complexnumber z, denoted by arg z. In these notes, the principal value of the argument of thecomplex number z, denoted by Argz, is defined to lie in the interval −π < Argz ≤ π.That is, Arg z is single-valued and is continuous at all points in the complex plane
1
excluding a branch cut along the negative real axis. In Appendix A, a detailed reviewof the properties of arg z and Arg z are provided.
The properties of Arg z determine the location of the branch cuts of the principalvalues of the logarithm the square root functions. The complex logarithm and gener-alized power functions are reviewed in Appendix B. If f(z) is expressible in terms ofthe logarithm the square root functions, then the definition of the principal value ofF (z) is not unique. However given a specific definition of F (z) in terms of the prin-cipal values of the logarithm the square root functions, the locations of the branchcuts of F (z) are inherited from that of Arg z and are thus uniquely determined.
2 The inverse trigonometric functions: arctan and
arccot
We begin by examining the solution to the equation
z = tanw =sinw
cosw=
1
i
(eiw − e−iw
eiw + e−iw
)
=1
i
(e2iw − 1
e2iw + 1
)
.
We now solve for e2iw,
iz =e2iw − 1
e2iw + 1=⇒ e2iw =
1 + iz
1− iz.
Taking the complex logarithm of both sides of the equation, we can solve for w,
w =1
2iln
(1 + iz
1− iz
)
.
The solution to z = tanw is w = arctan z. Hence,
arctan z =1
2iln
(1 + iz
1− iz
)
(1)
Since the complex logarithm is a multi-valued function, it follows that the arctangentfunction is also a multi-valued function. Using the definition of the multi-valuedcomplex logarithm,
arctan z =1
2iLn
∣∣∣∣
1 + iz
1− iz
∣∣∣∣+ 1
2
[
Arg
(1 + iz
1− iz
)
+ 2πn
]
, n = 0 , ±1 , ±2 , . . . , (2)
where Arg is the principal value of the argument function.Similarly,
z = cotw =cosw
sinw=
(i(eiw + e−iw
eiw − e−iw
)
=
(i(e2iw + 1
e2iw − 1
)
.
2
Again, we solve for e2iw,
−iz =e2iw + 1
e2iw − 1=⇒ e2iw =
iz − 1
iz + 1.
Taking the complex logarithm of both sides of the equation, we conclude that
w =1
2iln
(iz − 1
iz + 1
)
=1
2iln
(z + i
z − i
)
,
after multiplying numerator and denominator by −i to get a slightly more convenientform. The solution to z = cotw is w = arccotz. Hence,
arccotz =1
2iln
(z + i
z − i
)
(3)
Thus, the arccotangent function is a multivalued function,
arccotz =1
2iLn
∣∣∣∣
z + i
z − i
∣∣∣∣+ 1
2
[
Arg
(z + i
z − i
)
+ 2πn
]
, n = 0 , ±1 , ±2 , . . . , (4)
Using the definitions given by eqs. (1) and (3), the following relation is easilyderived:
arccot(z) = arctan
(1
z
)
. (5)
Note that eq. (5) can be used as the definition of the arccotangent function. Itis instructive to derive another relation between the arctangent and arccotangentfunctions. First, we first recall the property of the multi-valued complex logarithm,
ln(z1z2) = ln(z1) + ln(z2) , (6)
as a set equality [cf. eq. (B.14)]. It is convenient to define a new variable,
v =i− z
i+ z, =⇒ −1
v=
z + i
z − i. (7)
It follows that:
arctan z + arccot z =1
2i
[
ln v + ln
(
−1
v
)]
=1
2iln
(−v
v
)
=1
2iln(−1) .
Since ln(−1) = i(π + 2πn) for n = 0,±1,±2 . . ., we conclude that
arctan z + arccot z = 12π + πn , for n = 0,±1,±2, . . . (8)
Finally, we mention two equivalent forms for the multi-valued complex arctangentand arccotangent functions. Recall that the complex logarithm satisfies
ln
(z1z2
)
= ln z1 − ln z2 , (9)
3
where this equation is to be viewed as a set equality [cf. eq. (B.15). Thus, the multi-valued arctangent and arccotangent functions given in eqs. (1) and (5), respectively,are equivalent to
arctan z =1
2i
[
ln(1 + iz)− ln(1− iz)
]
, (10)
arccot z =1
2i
[
ln
(
1 +i
z
)
− ln
(
1− i
z
)]
, (11)
3 The principal values Arctan and Arccot
It is convenient to define principal values of the inverse trigonometric functions, whichare single-valued functions, which will necessarily exhibit a discontinuity across branchcuts in the complex plane. In Mathematica 8, the principal values of the complexarctangent and arccotangent functions, denoted by Arctan and Arccot respectively(with a capital A), are defined by employing the principal values of the complexlogarithms in eqs. (10) and (11),
Arctan z =1
2i
[
Ln(1 + iz)− Ln(1− iz)
]
, z 6= ±i (12)
and
Arccot z = Arctan
(1
z
)
=1
2i
[
Ln
(
1 +i
z
)
− Ln
(
1− i
z
)]
, z 6= ±i , z 6= 0 (13)
Note that the points z = ±i are excluded from the above definitions, as the arctangentand arccotangent are divergent at these two points. The definition of the principalvalue of the arccotangent given in eq. (13) is deficient in one respect since it is notwell-defined at z = 0. We shall address this problem shortly. One useful feature ofthe definitions eqs. (12) and (13) is that they satisfy:
Arctan(−z) = −Arctan z , Arccot(−z) = −Arccot z .
Because the principal value of the complex logarithm Ln does not satisfy eq. (9)in all regions of the complex plane, it follows that the definitions of the complexarctangent and arccotangent functions adopted by Mathematica 8 do not coincidewith alternative definitions that employ the principal value of the complex logarithmsin eqs. (1) and (4) [for further details, see Appendix C].
First, we shall identify the location of the branch cuts by identifying the linesof discontinuity of the principal values of the complex arctangent and arccotangent
4
functions in the complex plane. The principal value of the complex arctangent func-tion is single-valued for all complex values of z, excluding the two branch points atz 6= ±i. Moreover, the the principal-valued logarithms, Ln (1± iz) are discontinuousas z crosses the lines 1 ± iz < 0, respectively. We conclude that Arctan z must bediscontinuous when z = x+ iy crosses lines on the imaginary axis such that
x = 0 and −∞ < y < −1 and 1 < y < ∞ . (14)
These two lines that lie along the imaginary axis are the branch cuts of Arctanz. Notethat Arctan z is single-valued on the branch cut itself, since it inherits this propertyfrom the principal value of the complex logarithm.
Likewise, the principal value of the complex arccotangent function is single-valuedfor all complex z, excluding the branch points z 6= ±i. Moreover, the the principal-valued logarithms, Ln
(1± i
z
)are discontinuous as z crosses the lines 1 ± i
z< 0,
respectively. We conclude that Arccot z must be discontinuous when z = x + iycrosses the branch cut located on the imaginary axis such that
x = 0 and − 1 < y < 1 . (15)
In particular, due to the presence of the branch cut,
limx→0−
Arccot(x+ iy) 6= limx→0+
Arccot(x+ iy) , for −1 < y < 1 ,
for real values of x, where 0+ indicates that the limit is approached from positive realaxis and 0− indicates that the limit is approached from negative real axis. If z 6= 0,eq. (13) provides unique values for Arccot z for all z 6= ±i in the complex plane,including points on the branch cut. Using eq. (12), one can easily show that if z is anon-zero complex number infinitesimally close to 0, then
Arccot z =z→0 , z 6=0
12π , for Re z > 0 ,
12π , for Re z = 0 and Im z < 0 ,
−12π , for Re z < 0 ,
−12π , for Re z = 0 and Im z > 0 .
(16)
It is now apparent why z = 0 is problematical in eq. (13), since limz→0Arccot z isnot uniquely defined by eq. (16). If we wish to define a single-valued arccotangentfunction, then we must separately specify the value of Arccot(0). Mathematica 8supplements the definition of the principal value of the complex arccotangent givenin eq. (13) by declaring that
Arccot(0) = 12π . (17)
With the definitions given in eqs. (12), (13) and (17), Arctan z and Arccot zare single-valued functions in the entire complex plane, excluding the branch pointsz = ±i, and are continuous functions as long as the complex number z does not crossthe branch cuts specified in eqs. (14) and (15), respectively.
5
Having defined precisely the principal values of the complex arctangent and ar-ccotangent functions, let us check that they reduce to the conventional definitionswhen z is real. First consider the principal value of the real arctangent function,which satisfies,1
−12π ≤ Arctan x ≤ 1
2π , for −∞ ≤ x ≤ ∞ , (18)
where x is a real variable. The definition given by eq. (12) does reduce to the conven-tional definition of the principal value of the real-valued arctangent function when zis real. In particular, for real values of x,
Arctan x =1
2i
[
Ln(1 + ix)− Ln(1− ix)
]
= 12
[
Arg(1 + ix)− Arg(1− ix)
]
, (19)
after noting that Ln|1 + ix| = Ln|1− ix| = 12Ln(1 + x2). Geometrically, the quantity
Arg(1+ ix)−Arg(1− ix) is the angle between the complex numbers 1+ ix and 1− ixviewed as vectors lying in the complex plane. This angle varies between −π and πover the range −∞ < x < ∞. Moreover, the values ±π are achieved in the limit asx → ±∞, respectively. Hence, we conclude that the principal interval of the real-valued arctangent function is indeed given by eq. (18). For all possible values of xexcluding x = −∞, one can check that it is permissible to subtract the two principal-valued logarithms (or equivalently the two Arg functions) using eq. (9). In the caseof x → −∞, eq. (B.21) yields Arg(1 + ix) − Arg(1 − ix) → −π, corresponding toN− = −1 in the notation of eq. (A.21). Hence, an extra term appears when combiningthe two logarithms that is equal to 2πiN− = −2πi. The end result is,
Arctan(−∞) =1
2i[ln(−1)− 2πi] = −1
2π ,
as required. As a final check, we can use the results of Tables 1 and 2 given inAppendix A to conclude that Arg(a+ bi) = Arctan(b/a) for a > 0. Setting a = 1 andb = x then yields:
Arg (1 + ix) = Arctanx , Arg (1− ix) = Arctan(−x) = −Arctan x .
Subtracting these two results yields eq. (19).In contrast to the real arctangent function, there is no generally agreed definition
for the principal range of the real-valued arccotangent function. However, a growingconsensus among computer scientists has led to the following choice for the principalrange of the real-valued arccotangent function,
−12π < Arccotx ≤ 1
2π , for −∞ ≤ x ≤ ∞ , (20)
1In defining the principal value of the real arctangent, we follow the conventions of Keith B. Old-ham, Jan Myland and Jerome Spanier, An Atlas of Functions (Springer Science, New York, 2009),Chapter 35.
6
where x is a real variable. Note that the principal value of the arccotangent functiondoes not include the endpoint −1
2π [contrast this with eq. (18) for Arctan]. The
reason for this behavior is that Arccotx is discontinuous at x = 0, with
limx→0−
Arccot x = −12π , lim
x→0+Arccotx = 1
2π , (21)
as a consequence of eq. (16). In particular, eq. (20) corresponds to the convention inwhich Arccot(0) = 1
2π [cf. eq. (17)]. Thus, as x increases from negative to positive
values, Arccotx never reaches −12π but jumps discontinuously to 1
2π at x = 0.
Finally, we examine the the analog of eq. (8) for the corresponding principalvalues. Employing the Mathematica 8 definitions for the principal values of thecomplex arctangent and arccotangent functions, we find that
Arctan z +Arccot z =
12π , for Re z > 0 ,
12π , for Re z = 0 , and Im z > 1 or −1 < Im z ≤ 0 ,
−12π , for Re z < 0 ,
−12π , for Re z = 0 , and Im z < −1 or 0 < Im z < 1 .
(22)The derivation of this result will be given in Appendix D. In Mathematica, one canconfirm eq. (22) with many examples.
The relations between the single-valued and multi-valued functions can be sum-marized by:
arctan z = Arctan z + nπ , n = 0 , ±1 , ±2 , · · · ,arccot z = Arccot z + nπ , n = 0 , ±1 , ±2 , · · · .
Note that we can use these relations along with eq. (22) to confirm the result obtainedin eq. (8).
4 The inverse trigonometric functions: arcsin and
arccos
The arcsine function is the solution to the equation:
z = sinw =eiw − e−iw
2i.
Letting v ≡ eiw and multiplying the resulting equation by v yields the quadraticequation,
v2 − 2izv − 1 = 0 . (23)
The solution to eq. (23) is:v = iz + (1− z2)1/2 . (24)
7
Since z is a complex variable, (1− z2)1/2 is the complex square-root function. This isa multi-valued function with two possible values that differ by an overall minus sign.Hence, we do not explicitly write out the ± sign in eq. (24). To avoid ambiguity, weshall write
v = iz + (1− z2)1/2 = iz + e12 ln(1−z
2) = iz + e12 [Ln|1−z
2|+i arg(1−z2)]
= iz + |1− z2|1/2e i
2arg(1−z2) .
In particular, note that
ei
2arg(1−z2) = e
i
2Arg(1−z2)einπ = ±e
i
2Arg(1−z2) , for n = 0, 1 ,
which exhibits the two possible sign choices.By definition, v ≡ eiw, from which it follows that
w =1
iln v =
1
iln(
iz + |1− z2|1/2e i
2arg(1−z2)
)
.
The solution to z = sinw is w = arcsin z. Hence,
arcsin z =1
iln(
iz + |1− z2|1/2e i
2arg(1−z2)
)
The arccosine function is the solution to the equation:
z = cosw =eiw + e−iw
2.
Letting v ≡ eiw , we solve the equation
v +1
v= 2z .
Multiplying by v, one obtains a quadratic equation for v,
v2 − 2zv + 1 = 0 . (25)
The solution to eq. (25) is:v = z + (z2 − 1)1/2 .
Following the same steps as in the analysis of arcsine, we write
w = arccos z =1
iln v =
1
iln[z + (z2 − 1)1/2
], (26)
where (z2 − 1)1/2 is the multi-valued square root function. More explicitly,
arccos z =1
iln(
z + |z2 − 1|1/2e i
2arg(z2−1)
)
. (27)
8
It is sometimes more convenient to rewrite eq. (27) in a slightly different form. Recallthat
arg(z1z2) = arg z + arg z2 , (28)
as a set equality [cf. eq. (A.10). We now substitute z1 = z and z2 = −1 into eq. (28)and note that arg(−1) = π + 2πn (for n = 0,±1,±2, . . .) and arg z = arg z + 2πn asa set equality. It follows that
arg(−z) = π + arg z ,
as a set equality. Thus,
ei2 arg(z2−1) = eiπ/2 e
i2 arg(1−z
2) = iei2 arg(1−z
2) ,
and we can rewrite eq. (26) as follows:
arccos z =1
iln(
z + i√1− z2
)
, (29)
which is equivalent to the more explicit form,
arccos z =1
iln(
z + i|1− z2|1/2e i
2arg(1−z2)
)
The arcsine and arccosine functions are related in a very simple way. Usingeq. (24),
i
v=
i
iz +√1− z2
=i(−iz +
√1− z2)
(iz +√1− z2)(−iz +
√1− z2)
= z + i√1− z2 ,
which we recognize as the argument of the logarithm in the definition of the arccosine[cf. eq. (29)]. Using eq. (6), it follows that
arcsin z + arccos z =1
i
[
ln v + ln
(i
v
)]
=1
iln
(iv
v
)
=1
iln i .
Since ln i = i(12π + 2πn) for n = 0,±1,±2 . . ., we conclude that
arcsin z + arccos z = 12π + 2πn , for n = 0,±1,±2, . . . (30)
5 The principal values Arcsin and Arccos
In Mathematica 8, the principal value of the arcsine function is obtained by employingthe principal value of the logarithm and the principle value of the square-root function(which corresponds to employing the principal value of the argument). Thus,
Arcsin z =1
iLn
(
iz + |1− z2|1/2e i
2Arg(1−z2)
)
. (31)
9
It is convenient to introduce some notation for the the principle value of the square-root function. Consider the multivalued square root function, denoted by z1/2. Hence-forth, we shall employ the symbol
√z to denote the single-valued function,
√z =
√
|z| e 12Arg z , (32)
where√
|z| denotes the unique positive squared root of the real number |z|. In thisnotation, eq. (31) is rewritten as:
Arcsin z =1
iLn
(
iz +√1− z2
)
(33)
One noteworthy property of the principal value of the arcsine function is
Arcsin(−z) = −Arcsin z . (34)
To prove this result, it is convenient to define:
v = iz +√1− z2 ,
1
v=
1
iz +√1− z2
= −iz +√1− z2 . (35)
Then,
Arcsin z =1
iLn v , Arcsin(−z) =
1
iLn
(1
v
)
.
The second logarithm above can be simplified by making use of eq. (B.23),
Ln(1/z) =
{
−Ln(z) + 2πi , if z is real and negative ,
−Ln(z) , otherwise .(36)
In Appendix E, we prove that v can never be real and negative. Hence it follows fromeq. (36) that
Arcsin(−z) =1
iLn
(1
v
)
= −1
iLn v = −Arcsin z ,
as asserted in eq. (34).We now examine the principal value of the arcsine for real-valued arguments such
that −1 ≤ x ≤ 1. Setting z = x, where x is real and |x| ≤ 1,
Arcsin x =1
iLn
(
ix+√1− x2
)
=1
i
[
Ln∣∣∣ix+
√1− x2
∣∣∣ + iArg
(
ix+√1− x2
)]
= Arg(
ix+√1− x2
)
, for |x| ≤ 1 , (37)
since ix +√1− x2 is a complex number with magnitude equal to 1 when x is real
with |x| ≤ 1. Moreover, ix +√1− x2 lives either in the first or fourth quadrant of
the complex plane, since Re(ix+√1− x2) ≥ 0. It follows that:
−π
2≤ Arcsin x ≤ π
2, for |x| ≤ 1 .
10
In Mathematica 8, the principal value of the arccosine is defined by:
Arccos z = 12π − Arcsin z . (38)
We demonstrate below that this definition is equivalent to choosing the principalvalue of the complex logarithm and the principal value of the square root in eq. (29).That is,
Arccos z =1
iLn
(
z + i√1− z2
)
(39)
To verify that eq. (38) is a consequence of eq. (39), we employ the notation of eq. (35)to obtain:
Arcsin z +Arccos z =1
i
[
Ln|v|+ Ln
(1
|v|
)
+ iArg v + iArg
(i
v
)]
= Arg v +Arg
(i
v
)
. (40)
It is straightforward to check that:
Arg v +Arg
(i
v
)
= 12π , for Re v ≥ 0 .
However in Appendix F, we prove that Re v ≡ Re (iz+√1− z2) ≥ 0 for all complex
numbers z. Hence, eq. (40) yields:
Arcsin z +Arccos z = 12π ,
as claimed.We now examine the principal value of the arccosine for real-valued arguments
such that −1 ≤ x ≤ 1. Setting z = x, where x is real and |x| ≤ 1,
Arccosx =1
iLn
(
x+ i√1− x2
)
=1
i
[
Ln∣∣∣x+ i
√1− x2
∣∣∣+ iArg
(
x+ i√1− x2
)]
= Arg(
x+ i√1− x2
)
, for |x| ≤ 1 , (41)
since x + i√1− x2 is a complex number with magnitude equal to 1 when x is real
with |x| ≤ 1. Moreover, x + i√1− x2 lives either in the first or second quadrant of
the complex plane, since Im(x+ i√1− x2) ≥ 0. It follows that:
0 ≤ Arccosx ≤ π , for |x| ≤ 1 .
The principal value of the complex arcsine and arccosine functions are single-valued for all complex z. The choice of branch cuts for Arcsin z and Arccos z mustcoincide in light of eq. (38). Moreover, due to the standard branch cut of the principalvalue square root function,2 it follows that Arcsin z is discontinuous when z = x+ iy
2One can check that the branch cut of the Ln function in eq. (33) is never encountered for anyfinite value of z. For example, in the case of Arcsin z, the branch cut of Ln can only be reached ifiz +
√1− z2 is real and negative. But this never happens since if iz +
√1− z2 is real then z = iy
for some real value of y, in which case iz +√1− z2 = −y +
√
1 + y2 > 0.
11
crosses lines on the real axis such that3
y = 0 and −∞ < x < −1 and 1 < x < ∞ . (42)
These two lines comprise the branch cuts of Arcsin z and Arccos z; each branch cutends at a branch point located at x = −1 and x = 1, respectively (although thesquare root function is not divergent at these points).4
To obtain the relations between the single-valued and multi-valued functions, wefirst notice that the multi-valued nature of the logarithms imply that arcsin z can takeon the values Arcsinz+2πn and arccos z can take on the values Arccosz+2πn, wheren is any integer. However, we must also take into account the fact that (1 − z2)1/2
can take on two values, ±√1− z2. In particular,
arcsin z =1
iln(iz ±
√1− z2) =
1
iln
( −1
iz ∓√1− z2
)
=1
i
[
ln(−1)− ln(iz ∓√1− z2)
]
= −1
iln(iz ∓
√1− z2) + (2n+ 1)π ,
where n is any integer. Likewise,
arccos z =1
iln(z ± i
√1− z2) =
1
iln
(1
z ∓ i√1− z2
)
= −1
iln(z ∓ i
√1− z2) + 2πn ,
where n is any integer. Hence, it follows that
arcsin z = (−1)nArcsin z + nπ , n = 0 , ±1 , ±2 , · · · , (43)
arccos z = ±Arccos z + 2nπ , n = 0 , ±1 , ±2 , · · · , (44)
where either ±Arccos z can be employed to obtain a possible value of arccos z. Inparticular, the choice of n = 0 in eq. (44) implies that:
arccos z = − arccos z , (45)
which should be interpreted as a set equality. Note that one can use eqs. (43) and(44) along with eq. (38) to confirm the result obtained in eq. (30).
6 The inverse hyperbolic functions: arctanh and
arccoth
Consider the solution to the equation
z = tanhw =sinhw
coshw=
(ew − e−w
ew + e−w
)
=
(e2w − 1
e2w + 1
)
.
3Note that for real w, we have | sinw| ≤ 1 and | cosw| ≤ 1. Hence, for both the functionsw = Arcsin z and w = Arccos z, it is desirable to choose the branch cuts to lie outside the intervalon the real axis where |Re z| ≤ 1.
4The functions Arcsin z and Arccos z also possess a branch point at the point of infinity (whichis defined more precisely in footnote 5). This can be verified by demonstrating that Arcsin(1/z) andArccos(1/z) possess a branch point at z = 0. For further details, see e.g. Section 58 of ref 3.
12
We now solve for e2w,
z =e2w − 1
e2w + 1=⇒ e2w =
1 + z
1− z.
Taking the complex logarithm of both sides of the equation, we can solve for w,
w =1
2ln
(1 + z
1− z
)
.
The solution to z = tanhw is w = arctanhz. Hence,
arctanh z =1
2ln
(1 + z
1− z
)
(46)
Similarly, by considering the solution to the equation
z = cothw =coshw
sinhw=
(ew + e−w
ew − e−w
)
=
(e2w + 1
e2w − 1
)
.
we end up with:
arccothz =1
2ln
(z + 1
z − 1
)
(47)
The above results then yield:
arccoth(z) = arctanh
(1
z
)
,
as a set equality.Finally, we note the relation between the inverse trigonometric and the inverse
hyperbolic functions:
arctanh z = i arctan(−iz) ,
arccoth z = i arccot(iz) .
As in the discussion at the end of Section 1, one can rewrite eqs. (46) and (47) inan equivalent form:
arctanh z = 12[ln(1 + z)− ln(1− z)] , (48)
arccoth z = 12
[
ln
(
1 +1
z
)
− ln
(
1− 1
z
)]
. (49)
13
7 The principal values Arctanh and Arccoth
Mathematica 8 defines the principal values of the inverse hyperbolic tangent andinverse hyperbolic cotangent, Arctanh and Arccoth, by employing the principal valueof the complex logarithms in eqs. (48) and (49). We can define the principal valueof the inverse hyperbolic tangent function by employing the principal value of thelogarithm,
Arctanh z = 12[Ln(1 + z)− Ln(1− z)] (50)
and
Arccoth z = Arctanh
(1
z
)
= 12
[
Ln
(
1 +1
z
)
− Ln
(
1− 1
z
)]
(51)
Note that the branch points at z = ±1 are excluded from the above definitions, asArctanh z and Arccoth z are divergent at these two points. The definition of theprincipal value of the inverse hyperbolic cotangent given in eq. (51) is deficient in onerespect since it is not well-defined at z = 0. For this special case, Mathematica 8defines
Arccoth(0) = 12iπ . (52)
Of course, this discussion parallels that of Section 2. Moreover, alternative def-initions of Arctanh z and Arccoth z analogous to those defined in Appendix C forthe corresponding inverse trigonometric functions can be found in ref. 4. There isno need to repeat the analysis of Section 2 since a comparison of eqs. (12) and (13)with eqs. (50) and (51) shows that the inverse trigonometric and inverse hyperbolictangent and cotangent functions are related by:
Arctanh z = iArctan(−iz) , (53)
Arccoth z = iArccot(iz) . (54)
Using these results, all other properties of the inverse hyperbolic tangent and cotan-gent functions can be easily derived from the properties of the corresponding arctan-gent and arccotangent functions.
For example the branch cuts of these functions are easily obtained from eqs. (14)and (15). Arctanh z is discontinuous when z = x+ iy crosses the branch cuts locatedon the real axis such that5
y = 0 and −∞ < x < −1 and 1 < x < ∞ . (55)
Arccoth z is discontinuous when z = x + iy crosses the branch cuts located on thereal axis such that
y = 0 and − 1 < x < 1 . (56)
5Note that for real w, we have | tanhw| ≤ 1 and | cothw| ≥ 1. Hence, for w = Arctanh z itis desirable to choose the branch cut to lie outside the interval on the real axis where |Re z| ≤ 1.Likewise, for w = Arccothz it is desirable to choose the branch cut to lie outside the interval on thereal axis where |Re z| ≥ 1.
14
The relations between the single-valued and multi-valued functions can be sum-marized by:
arctanhz = Arctanh z + inπ , n = 0 , ±1 , ±2 , · · · ,arccoth z = Arccoth z + inπ , n = 0 , ±1 , ±2 , · · · .
8 The inverse hyperbolic functions: arcsinh and
arccosh
The inverse hyperbolic sine function is the solution to the equation:
z = sinhw =ew − e−w
2.
Letting v ≡ ew , we solve the equation
v − 1
v= 2z .
Multiplying by v, one obtains a quadratic equation for v,
v2 − 2zv − 1 = 0 . (57)
The solution to eq. (57) is:v = z + (1 + z2)1/2 . (58)
Since z is a complex variable, (1+ z2)1/2 is the complex square-root function. This isa multi-valued function with two possible values that differ by an overall minus sign.Hence, we do not explicitly write out the ± sign in eq. (58). To avoid ambiguity, weshall write
v = z + (1 + z2)1/2 = z + e12 ln(1+z2) = z + e
12 [Ln|1+z2|+i arg(1+z2)]
= z + |1 + z2|1/2e i
2arg(1+z2) .
By definition, v ≡ ew, from which it follows that
w = ln v = ln(
z + |1 + z2|1/2e i
2arg(1+z2)
)
.
The solution to z = sinhw is w = arcsinhz. Hence,
arcsinh z = ln(
z + |1 + z2|1/2e i
2arg(1+z2)
)
(59)
15
The inverse hyperbolic cosine function is the solution to the equation:
z = cosw =ew + e−w
2.
Letting v ≡ ew , we solve the equation
v +1
v= 2z .
Multiplying by v, one obtains a quadratic equation for v,
v2 − 2zv + 1 = 0 . (60)
The solution to eq. (60) is:v = z + (z2 − 1)1/2 .
Following the same steps as in the analysis of inverse hyperbolic sine function, wewrite
w = arccosh z = ln v = ln[z + (z2 − 1)1/2
], (61)
where (z2 − 1)1/2 is the multi-valued square root function. More explicitly,
arccosh z = ln(
z + |z2 − 1|1/2e i
2arg(z2−1)
)
The multi-valued square root function satisfies:
(z2 − 1)1/2 = (z + 1)1/2(z − 1)1/2 .
Hence, an equivalent form for the multi-valued inverse hyperbolic cosine function is:
arccosh z = ln[z + (z + 1)1/2(z − 1)1/2
],
or equivalently,
arccosh z = ln(
z + |z2 − 1|1/2e i
2arg(z+1)e
i
2arg(z−1)
)
. (62)
Finally, we note the relations between the inverse trigonometric and the inversehyperbolic functions:
arcsinh z = i arcsin(−iz) , (63)
arccosh z = ± i arccos z , (64)
where the equalities in eqs. (63) and (64) are interpreted as set equalities for themulti-valued functions. The ± in eq. (64) indicates that both signs are employedin determining the members of the set of all possible arccosh z values. In derivingeq. (64), we have employed eqs. (26) and (61). In particular, the origin of the twopossible signs in eq. (64) is a consequence of eq. (45) [and its hyperbolic analog,eq. (73)].
16
9 The principal values Arcsinh and Arccosh
The principal value of the inverse hyperbolic sine function, Arcsinh z, is defined byMathematica 8 by replacing the complex logarithm and argument functions of eq. (59)by their principal values. That is,
Arcsinh z = Ln(
z +√1 + z2
)
(65)
For the principal value of the inverse hyperbolic cosine function Arccoshz, Mathemat-ica 8 chooses eq. (62) with the complex logarithm and argument functions replacedby their principal values. That is,
Arccosh z = Ln(z +
√z + 1
√z − 1
)(66)
In eqs. (65) and (66), the principal values of the square root functions are employedfollowing the notation of eq. (32).
The relation between the principal values of the inverse trigonometric and theinverse hyperbolic sine functions is given by
Arcsinh z = iArcsin(−iz) , (67)
as one might expect in light of eq. (63). A comparison of eqs. (39) and (66) revealsthat
Arccosh z =
{
iArccos z , for either Im z > 0 or for Im z = 0 and Re z ≤ 1 ,
−iArccos z , for either Im z < 0 or for Im z = 0 and Re z ≥ 1 .
(68)The existence of two possible signs in eq. (68) is not surprising in light of the ± thatappears in eq. (64). Note that either choice of sign is valid in the case of Im z = 0 andRe z = 1, since for this special point, Arccosh(1) = Arccos(1) = 0 . For a derivationof eq. (68), see Appendix F.
The principal value of the inverse hyperbolic sine and cosine functions are single-valued for all complex z. Moreover, due to the branch cut of the principal valuesquare root function,6 it follows that Arcsinh z is discontinuous when z = x + iycrosses lines on the imaginary axis such that
x = 0 and −∞ < y < −1 and 1 < y < ∞ . (69)
These two lines comprise the branch cuts of Arcsinh z, and each branch cut endsat a branch point located at z = −i and z = i, respectively, due to the square root
6One can check that the branch cut of the Ln function in eq. (65) is never encountered for anyvalue of z. In particular, the branch cut of Ln can only be reached if z+
√1 + z2 is real and negative.
But this never happens since if z +√1 + z2 is real then z is also real. But for any real value of z,
we have z +√1 + z2 > 0.
17
function in eq. (65), although the square root function is not divergent at these points.The function Arcsinh z also possesses a branch point at the point of infinity, whichcan be verified by examining the behavior of Arcsinh(1/z) at the point z = 0.7
The branch cut for Arccosh z derives from the standard branch cuts of the squareroot function and the branch cut of the complex logarithm. In particular, for real zsatisfying |z| < 1, we have a branch cut due to (z+1)1/2(z− 1)1/2, whereas for real zsatisfying −∞ < z ≤ −1, the branch cut of the complex logarithm takes over. Hence,it follows that Arccosh z is discontinuous when z = x + iy crosses lines on the realaxis such that8
y = 0 and −∞ < x < 1 . (70)
In particular, there are branch points at z = ±1 due to the square root functions ineq. (66) and a branch point at the point of infinity due to the logarithm [cf. footnote 5].As a result, eq. (70) actually represents two branch cuts made up of a branch cutfrom z = 1 to z = −1 followed by a second branch cut from z = −1 to the point ofinfinity.9
The relations between the single-valued and multi-valued functions can be ob-tained by following the same steps used to derive eqs. (43) and (44). Alternatively,we can make use of these results along with those of eqs. (63), (64), (67) and (68).The end result is:
arcsinh z = (−1)nArcsinh z + inπ , n = 0 , ±1 , ±2 , · · · , (71)
arccosh z = ±Arccosh z + 2inπ , n = 0 , ±1 , ±2 , · · · , (72)
where either ±Arccosh z can be employed to obtain a possible value of arccosh z. Inparticular, the choice of n = 0 in eq. (72) implies that:
arccosh z = −arccosh z , (73)
which should be interpreted as a set equality.This completes our survey of the multi-valued complex inverse trigonometric and
hyperbolic functions and their single-valued principal values.
7In the complex plane, the behavior of the complex function F (z) at the point of infinity, z = ∞,corresponds to the behavior of F (1/z) at the origin of the complex plane, z = 0 [cf. footnote 3].Since the argument of the complex number 0 is undefined, the argument of the point of infinityis likewise undefined. This means that the point of infinity (sometimes called complex infinity)actually corresponds to |z| = ∞, independently of the direction in which infinity is approached inthe complex plane. Geometrically, the complex plane plus the point of infinity can be mapped ontoa surface of a sphere by stereographic projection. Place the sphere on top of the complex plane suchthat the origin of the complex plane coincides with the south pole. Consider a straight line from anycomplex number in the complex plane to the north pole. Before it reaches the north pole, this lineintersects the surface of the sphere at a unique point. Thus, every complex number in the complexplane is uniquely associated with a point on the surface of the sphere. In particular, the north poleitself corresponds to complex infinity. For further details, see Chapter 5 of ref. 3.
8Note that for real w, we have coshw ≥ 1. Hence, for w = Arccoshz it is desirable to choose thebranch cut to lie outside the interval on the real axis where Re z ≥ 1.
9Given that the branch cuts of Arccoshz and iArccosz are different, it is not surprising that therelation Arccosh z = iArccos z cannot be respected for all complex numbers z.
18
APPENDIX A: The argument of a complex number
In this Appendix, we examine the argument of a non-zero complex number z. Anycomplex number can be written as, we write:
z = x+ iy = r(cos θ + i sin θ) = reiθ , (A.1)
where x = Re z and y = Im z are real numbers. The complex conjugate of z, denotedby z, is given by
z = x− iy = re−iθ .
The argument of z is denoted by θ, which is measured in radians. However, thereis an ambiguity in definition of the argument. The problem is that
sin(θ + 2π) = sin θ , cos(θ + 2π) = cos θ ,
since the sine and the cosine are periodic functions of θ with period 2π. Thus θis defined only up to an additive integer multiple of 2π. It is common practice toestablish a convention in which θ is defined to lie within an interval of length 2π. Thisdefines the so-called principal value of the argument, which we denote by θ ≡ Arg z(note the capital A). The most common convention, which we adopt in these notes,is to define the single-valued function Arg z such that:
−π < Arg z ≤ π . (A.2)
In many applications, it is convenient to define a multi-valued argument function,
arg z ≡ Arg z + 2πn = θ + 2πn , n = 0,±1,±2,±3, . . . . (A.3)
This is a multi-valued function because for a given complex number z, the numberarg z represents an infinite number of possible values.
A.1. The principal value of the argument function
Any non-zero complex number z can be written in polar form
z = |z|ei arg z , (A.4)
where arg z is a multi-valued function defined in eq. (A.3) It is convenient to have anexplicit formula for Arg z in terms of arg z. First, we introduce some notation: [x]means the largest integer less than or equal to the real number x. That is, [x] is theunique integer that satisfies the inequality
x− 1 < [x] ≤ x , for real x and integer [x] . (A.5)
19
For example, [1.5] = [1] = 1 and [−0.5] = −1. With this notation, one can writeArg z in terms of arg z as follows:
Arg z = arg z + 2π
[1
2− arg z
2π
]
, (A.6)
where [ ] denotes the bracket (or greatest integer) function introduced above. It isstraightforward to check that Arg z as defined by eq. (A.6) does indeed fall inside theprincipal interval, −π < θ ≤ π.
A more useful equation for Arg z can be obtained as follows. Using the polarrepresentation of z = x + iy given in eq. (A.1), it follows that x = r cos θ andy = r sin θ. From these two results, one easily derives,
|z| = r =√
x2 + y2 , tan θ =y
x. (A.7)
We identify θ = Arg z in the convention where −π < θ ≤ π. In light of eq. (A.7), itis tempting to identify Arg z with arctan(y/x). However, the real function arctan xis a multi-valued function for real values of x. It is conventional to introduce a single-valued real arctangent function, called the principal value of the arctangent, whichis denoted by Arctan x and satisfies −1
2π ≤ Arctan x ≤ 1
2π [cf. eq. (18)]. Since
−π < Arg z ≤ π, it follows that Arg z cannot be identified with arctan(y/x) in allregions of the complex plane. The correct relation between these two quantities iseasily ascertained by considering the four quadrants of the complex plane separately.The quadrants of the complex plane (called regions I, II, III and IV) are illustratedin the figure below:
y
x
III
III IV
The principal value of the argument of z = x + iy in terms of its real part xand imaginary part y is given in Table 1, assuming that z lies within one of the fourquadrants of the complex plane. Note that Arg z = Arctan(y/x) is valid only inquadrants I and IV. If z lies within quadrants II or III, one must add or subtract πin order to ensure that 1
2π < Arg z < π or −π < Arg z < −1
2π, respectively.10
10For finite non-zero values of y/x, the principal value of the arctangent function lies inside theinterval 0 < Arctan(y/x) < 1
2π if y/x > 0 and − 1
2π < Arctan(y/x) < 0 if y/x < 0. For completeness,
we note that
Arctan(y/x) =
0 , if y = 0 and x 6= 0 ,1
2π , if x = 0 and y > 0 ,
− 1
2π , if x = 0 and y < 0 ,
undefined , if x = y = 0 .
20
Table 1: Formulae for the argument of a complex number z = x+ iy.
Quadrant Sign of x and y Arg z
I x > 0 , y > 0 Arctan(y/x)
II x < 0 , y > 0 π +Arctan(y/x)
III x < 0 , y < 0 −π +Arctan(y/x)
IV x > 0 , y < 0 Arctan(y/x)
Table 2: Formulae for the argument of a complex number z = x+iy when z is real or pureimaginary. By convention, the principal value of the argument satisfies −π < Arg z ≤ π.
Quadrant border type of complex number z Conditions on x and y Arg z
IV/I real and positive x > 0 , y = 0 0
I/II pure imaginary with Im z > 0 x = 0 , y > 0 12π
II/III real and negative x < 0 , y = 0 π
III/IV pure imaginary with Im z < 0 x = 0 , y < 0 −12π
origin zero x = y = 0 undefined
Cases where z lies on the border between two adjacent quadrants are consideredseparately in Table 2. In particular, note that the argument of zero is undefined.Since z = 0 if and only if |z| = 0, eq. (A.4) remains valid despite the fact that arg 0 isnot defined. When studying the properties of arg z and Arg z below, we shall alwaysassume implicitly that z 6= 0.
A.2. Properties of the multi-valued argument function
We can view a multi-valued function f(z) evaluated at z as a set of values, where eachelement of the set corresponds to a different choice of some integer n. For example,given the multi-valued function arg z whose principal value is Arg z ≡ θ, then arg zconsists of the set of values:
arg z = {θ , θ + 2π , θ − 2π , θ + 4π , θ − 4π , · · · } . (A.8)
Given two multi-valued functions, e.g., f(z) = F (z) + 2πn and g(z) = G(z) + 2πn,where F (z) and G(z) are the principal values of f(z) and g(z) respectively, then
21
f(z) = g(z) if and only if for each point z, the corresponding set of values of f(z)and g(z) precisely coincide:
{F (z) , F (z) + 2π , F (z)− 2π , · · · } = {G(z) , G(z) + 2π , G(z)− 2π , · · · } .(A.9)
Sometimes, one refers to the equation f(z) = g(z) as a set equality since all theelements of the two sets in eq. (A.9) must coincide. We add two additional rules tothe concept of set equality. First, the ordering of terms within the set is unimportant.Second, we only care about the distinct elements of each set. That is, if our list ofset elements has repeated entries, we omit all duplicate elements.
To see how the set equality of two multi-valued functions works, let us considerthe multi-valued function arg z. One can prove that:
arg(z1z2) = arg z1 + arg z2 , (A.10)
arg
(z1z2
)
= arg z1 − arg z2 . (A.11)
arg
(1
z
)
= arg z = − arg z . (A.12)
To prove eq. (A.10), consider z1 = |z1|ei arg z1 and z2 = |z2|ei arg z2 . The arguments ofthese two complex numbers are: arg z1 = Arg z1 + 2πn1 and arg z2 = Arg z2 + 2πn2,where n1 and n2 are arbitrary integers. [One can also write arg z1 and arg z2 in setnotation as in eq. (A.8).] Thus, one can also write z1 = |z1|eiArg z1 and z2 = |z2|eiArg z2 ,since e2πin = 1 for any integer n. It then follows that
z1z2 = |z1z2|ei(Arg z1+Arg z2) ,
where we have used |z1||z2| = |z1z2|. Thus, arg(z1z2) = Arg z1 + Arg z2 + 2πn12,where n12 is also an arbitrary integer. Therefore, we have established that:
arg z1 + arg z2 = Arg z1 +Arg z2 + 2π(n1 + n2) ,
arg(z1z2) = Arg z1 +Arg z2 + 2πn12 ,
where n1, n2 and n12 are arbitrary integers. Thus, arg z1 + arg z2 and arg(z1z2)coincide as sets, and so eq. (A.10) is confirmed. One can easily prove eqs. (A.11) and(A.12) by a similar method. In particular, if one writes z = |z|ei arg z and employs thedefinition of the complex conjugate (which yields |z| = |z| and z = |z|e−i arg z), thenit follows that arg(1/z) = arg z = − arg z. As an instructive example, consider thelast relation in the case of z = −1. It then follows that
arg(−1) = − arg(−1) ,
as a set equality. This is not paradoxical, since the sets,
arg(−1) = {±π , ±3π , ±5π , . . .} and − arg(−1) = {∓π , ∓3π , ∓5π , . . .} ,
22
coincide, as they possess precisely the same list of elements.Now, for a little surprise:
arg z2 6= 2 arg z . (A.13)
To see why this statement is surprising, consider the following false proof. Useeq. (A.10) with z1 = z2 = z to derive:
arg z2 = arg z + arg z?= 2 arg z , [FALSE!!]. (A.14)
The false step is the one indicated by the symbol?= above. Given z = |z|ei arg z, one
finds that z2 = |z|2e2i(Arg z+2πn) = |z|2e2iArg z, and so the possible values of arg(z2)are:
arg(z2) = {2Arg z , 2Arg z + 2π , 2Arg z − 2π , 2Arg z + 4π , 2Arg z − 4π , · · · } ,
whereas the possible values of 2 arg z are:
2 arg z = {2Arg z , 2(Arg z + 2π) , 2(Arg z − 2π) , 2(Arg z + 4π) , · · · }
= {2Arg z , 2Arg z + 4π , 2Arg z − 4π , 2Arg z + 8π , 2Arg z − 8π , · · · } .
Thus, 2 arg z is a subset of arg(z2), but half the elements of arg(z2) are missing from2 arg z. These are therefore unequal sets, as indicated by eq. (A.13). Now, you shouldbe able to see what is wrong with the statement:
arg z + arg z?= 2 arg z . (A.15)
When you add arg z as a set to itself, the element you choose from the first arg zneed not be the same as the element you choose from the second arg z. In contrast,2 arg z means take the set arg z and multiply each element by two. The end result isthat 2 arg z contains only half the elements of arg z+arg z as shown above. Similarly,for any non-negative integer n = 2, 3, 4, . . .,
arg zn = arg z + arg z + · · · arg z︸ ︷︷ ︸
n
6= n arg z . (A.16)
In light of eq. (A.12), if we replace z with z above, we obtain the following general-ization of eq. (A.16),
arg zn 6= n arg z , for any integer n 6= 0 , ±1 . (A.17)
Here is one more example of an incorrect proof. Consider eq. (A.11) with z1 =z2 ≡ z. Then, you might be tempted to write:
arg(z
z
)
= arg(1) = arg z − arg z?= 0 .
23
This is clearly wrong since arg(1) = 2πn, where n is the set of integers. Again, theerror occurs with the step:
arg z − arg z?= 0 . (A.18)
The fallacy of this statement is the same as above. When you subtract arg z as a setfrom itself, the element you choose from the first arg z need not be the same as theelement you choose from the second arg z.
A.3. Properties of the principal value of the argument
The properties of the principal value Arg z are not as simple as those given ineqs. (A.10)–(A.12), since the range of Arg z is restricted to lie within the princi-pal range −π < Arg z ≤ π. Instead, the following relations are satisfied:
Arg (z1z2) = Arg z1 +Arg z2 + 2πN+ , (A.19)
Arg (z1/z2) = Arg z1 − Arg z2 + 2πN− , (A.20)
where the integers N± are determined as follows:
N± =
−1 , if Arg z1 ± Arg z2 > π ,
0 , if −π < Arg z1 ± Arg z2 ≤ π ,
1 , if Arg z1 ± Arg z2 ≤ −π .
(A.21)
If we set z1 = 1 in eq. (A.20), we find that
Arg(1/z) = Arg z =
{
Arg z , if Im z = 0 and z 6= 0 ,
−Arg z , if Im z 6= 0 .(A.22)
Note that for z real, both 1/z and z are also real so that in this case z = z andArg(1/z) = Arg z = Arg z.
Finally,
Arg(zn) = nArg z + 2πNn , (A.23)
where the integer Nn is given by:
Nn =
[1
2− n
2πArg z
]
, (A.24)
and [ ] is the greatest integer bracket function introduced in eq. (A.5). It is straight-forward to verify eqs. (A.19)–(A.22) and eq. (A.23). These formulae follow from thecorresponding properties of arg z, taking into account the requirement that Arg zmust lie within the principal interval, −π < θ ≤ π.
24
APPENDIX B: The complex logarithm and its prin-
cipal value
B.1. Definition of the complex logarithm
In order to define the complex logarithm, one must solve the complex equation:
z = ew , (B.1)
for w, where z is any non-zero complex number. If we write w = u+ iv, then eq. (B.1)can be written as
eueiv = |z|ei arg z . (B.2)
Eq. (B.2) implies that:|z| = eu , v = arg z .
The equation |z| = eu is a real equation, so we can write u = ln |z|, where ln |z| is theordinary logarithm evaluated with positive real number arguments. Thus,
w = u+ iv = ln |z|+ i arg z = ln |z|+ i(Arg z + 2πn) , n = 0 , ±1 , ±2 , ±3 , . . .
(B.3)
We call w the complex logarithm and write w = ln z. This is a somewhat awkwardnotation since in eq. (B.3) we have already used the symbol ln for the real logarithm.We shall finesse this notational quandary by denoting the real logarithm in eq. (B.3)by the symbol Ln. That is, Ln|z| shall denote the ordinary real logarithm of |z|. Withthis notational convention, we rewrite eq. (B.3) as:
ln z = Ln|z| + i arg z = Ln|z|+ i(Arg z + 2πn) , n = 0 , ±1 , ±2 , ±3 , . . . (B.4)
for any non-zero complex number z.Clearly, ln z is a multi-valued function (as its value depends on the integer n). It
is useful to define a single-valued function complex function, Lnz, called the principalvalue of ln z as follows:
Ln z = Ln |z| + iArg z , −π < Arg z ≤ π , (B.5)
which extends the definition of Ln z to the entire complex plane (excluding the origin,z = 0, where the logarithmic function is singular). In particular, eq. (B.5) impliesthat Ln(−1) = iπ. Note that for real positive z, we have Arg z = 0, so that eq. (B.5)simply reduces to the usual real logarithmic function in this limit. The relationbetween ln z and its principal value is simple:
ln z = Ln z + 2πin , n = 0 , ±1 , ±2 , ±3 , . . . .
25
B.2. Properties of the real and complex logarithm
The properties of the real logarithmic function (whose argument is a positive realnumber) are well known:
elnx = x , (B.6)
ln(ea) = a , (B.7)
ln(xy) = ln(x) + ln(y) , (B.8)
ln
(x
y
)
= ln(x)− ln(y) , (B.9)
ln
(1
x
)
= − ln(x) , (B.10)
for positive real numbers x and y and arbitrary real number a. Repeated use ofeqs. (B.8) and (B.9) then yields
ln xn = n ln x , for arbitrary integer n . (B.11)
We now consider which of the properties given in eqs. (B.6)–(B.11) apply to thecomplex logarithm. Since we have defined the multi-value function ln z and the single-valued function Ln z, we should examine the properties of both these functions.We begin with the multi-valued function ln z. First, we examine eq. (B.6). Usingeq. (B.4), it follows that:
eln z = eLn|z|eiArg ze2πin = |z|eiArg z = z . (B.12)
Thus, eq. (B.6) is satisfied. Next, we examine eq. (B.7) for z = x+ iy:
ln(ez) = Ln|ez|+ i(arg ez) = Ln(ex) + i(y + 2πk) = x+ iy + 2πik = z + 2πik ,
where k is an arbitrary integer. In deriving this result, we used the fact that ez = exeiy,which implies that arg(ez) = y + 2πk.11 Thus,
ln(ez) = z + 2πik 6= z , unless k = 0 . (B.13)
This is not surprising, since ln(ez) is a multi-valued function, which cannot be equalto the single-valued function z. Indeed eq. (B.7) is false for the multi-valued complexlogarithm.
As a check, let us compute ln(eln z) in two different ways. First, using eq. (B.12),it follows that ln(eln z) = ln z. Second, using eq. (B.13), ln(eln z) = ln z + 2πik. This
11Note that Arg ez = y + 2πN , where N is chosen such that −π < y + 2πN ≤ π. Moreover,eq. (A.3) implies that arg ez = Arg ez + 2πn, where n = 0,±1,±2, . . .. Hence, arg(ez) = y + 2πk,where k = n+N is still some integer.
26
seems to imply that ln z = ln z + 2πik. In fact, the latter is completely valid as a set
equality in light of eq. (B.4).We now consider the properties exhibited in eqs. (B.8)–(B.10). Using the defi-
nition of the multi-valued complex logarithms and the properties of arg z given ineqs. (A.10)–(A.12), it follows that eqs. (B.8)–(B.11) are satisfied as set equalities:
ln(z1z2) = ln z1 + ln z2 , (B.14)
ln
(z1z2
)
= ln z1 − ln z2 . (B.15)
ln
(1
z
)
= − ln z . (B.16)
However, one must be careful in employing these results. One should not make the
mistake of writing, for example, ln z + ln z?= 2 ln z or ln z − ln z
?= 0. Both these
latter statements are false for the same reasons that eqs. (A.14) and (A.18) are not
identities under set equality. In general, the multi-valued complex logarithm does notsatisfy eq. (B.11) as a set equality. In particular,
ln zn 6= n ln z , for integer n 6= 0 , ±1 , (B.17)
as a consequence of eq. (A.17).We next examine the properties of the single-valued function Ln z. Again, we
examine the six properties given by eqs. (B.6)–(B.11). First, eq. (B.6) is triviallysatisfied since
eLn z = eLn|z|eiArg z = |z|eiArg z = z . (B.18)
However, eq. (B.7) is generally false. In particular, for z = x+ iy
Ln(ez) = Ln |ez|+ i(Arg ez) = Ln(ex) + i(Arg eiy) = x+ iArg (eiy)
= x+ i arg(eiy) + 2πi
[1
2− arg(eiy)
2π
]
= x+ iy + 2πi
[1
2− y
2π
]
= z + 2πi
[1
2− Im z
2π
]
, (B.19)
after using eq. (A.6), where [ ] is the greatest integer bracket function defined ineq. (A.5). Thus, eq. (B.7) is satisfied only when −π < y ≤ π. For values of y outsidethe principal interval, eq. (B.7) contains an additive correction term as shown ineq. (B.19).
As a check, let us compute Ln(eLnz) in two different ways. First, using eq. (B.18),it follows that Ln(eLn z) = Ln z. Second, using eq. (B.19),
Ln(eLn z) = ln z + 2πi
[1
2− Im Ln z
2π
]
= Ln z + 2πi
[1
2− Arg z
2π
]
= Ln z ,
27
where we have used Im Ln z = Arg z [see eq. (B.5)]. In the last step, we noted that
0 ≤ 1
2− Arg(z)
2π< 1 ,
due to eq. (A.2), which implies that the integer part of 12−Arg z/(2π) is zero. Thus,
the two computations agree.We now consider the properties exhibited in eqs. (B.8)–(B.11). Ln z may not
satisfy any of these properties due to the fact that the principal value of the complexlogarithm must lie in the interval −π < Im Ln z ≤ π. Using the results of eqs. (A.19)–(A.24), it follows that
Ln (z1z2) = Ln z1 + Ln z2 + 2πiN+ , (B.20)
Ln (z1/z2) = Ln z1 − Ln z2 + 2πiN− , (B.21)
Ln(zn) = nLn z + 2πiNn (integer n) , (B.22)
where the integers N± = −1, 0 or +1 and Nn are determined by eqs. (A.21) and(A.24), respectively, and
Ln(1/z) =
{
−Ln(z) + 2πi , if z is real and negative ,
−Ln(z) , otherwise .(B.23)
Note that eq. (B.8) is satisfied if Re z1 > 0 and Re z2 > 0 (in which case N+ = 0). Inother cases, N+ 6= 0 and eq. (B.8) fails. Similar considerations also apply to eqs. (B.9)and (B.10). For example, eq. (B.10) is satisfied by Ln z unless Arg z = π (equivalentlyfor negative real values of z), as indicated by eq. (B.23). In particular, one may useeq. (B.22) to verify that:
Ln[(−1)−1] = −Ln(−1) + 2πi = −πi+ 2πi = πi = Ln(−1) ,
as expected, since (−1)−1 = −1.
B.3. Properties of the generalized power function
The generalized complex power function is defined via the following equation:
w = zc = ec ln z , z 6= 0 . (B.24)
Having defined the multi-valued complex power function, we are now able to computeln(zc) for arbitrary complex number c and complex variable z,
ln(zc) = ln(ec ln z) = ln(ec (Ln z+2πim)) = ln(ecLn ze2πimc)
= ln(ecLn z) + ln(e2πimc) = c (Ln z + 2πim) + 2πik
= c ln z + 2πik = c
(
ln z +2πik
c
)
, (B.25)
28
where k and m are arbitrary integers. Thus, ln(zc) = c ln z in the sense of set equality(in which case the sets corresponding to ln z and ln z + 2πik/c coincide) if and onlyif k/c is an integer for all values of k. The only possible way to satisfy this latterrequirement is to take c = 1/n, where n is an integer.
We can define a single-valued power function by selecting the principal value ofln z in eq. (B.24). Consequently, the principal value of zc is defined by
Zc = ecLn z , z 6= 0 .
For a lack of a better notation, I will indicate the principal value by capitalizing thevariable Z as above. The principal value definition of zc can lead to some unexpectedresults. For example, consider the principal value of the cube root function w =Z1/3 = eLn(z)/3. Then, for z = −1, the principal value of
3√−1 = eLn(−1)/3 = eπi/3 = 1
2
(
1 + i√3)
.
This may have surprised you, if you were expecting that 3√−1 = −1. To obtain the
latter result would require a different choice of the principal interval in the definitionof the principal value of z1/3.
Employing the principal value for the complex logarithm and complex power func-tion, it follows that
Ln(Zc) = Ln(ecLn z) = cLn z + 2πiNc , (B.26)
after using eq. (B.19), where Nc is an integer determined by
Nc ≡[1
2− Im (cLn z)
2π
]
, (B.27)
and [ ] is the greatest integer bracket function defined in eq. (A.5). Nc can beevaluated by noting that:
Im (cLn z) = Im {c (Ln|z|+ iArg z)} = Arg z Re c+ Ln|z| Im c .
Note that if c = n where n is an integer, then eq. (B.26) simply reduces to eq. (B.22),as expected.
The properties of the generalized power function and its principal value followfrom the corresponding properties of the complex logarithm derived in AppendixB.2. For example, the single-valued power function satisfies:
ZaZb = eaLn zebLn z = e(a+b)Ln z = Za+b , (B.28)
Za
Zb=
eaLn z
ebLn z= e(a−b)Ln z = Za−b , (B.29)
ZaZ−a = eaLn ze−aLn z = 1 . (B.30)
29
Setting a = b in eq. (B.29) yields Z0 = 1 (for Z 6= 0) as expected.In contrast,
(Zc)b = (ecLn z)b = ebLn(ecLn z) = eb(cLn z+2πiNc) = ebcLn z e2πibNc = Zcb e2πibNc ,
where Nc is an integer determined by eq. (B.27). As an example, if z = b = c = i,eq. (B.27) gives Nc = 0, which yields the principal value of (ii)i = ii·i = i−1 = −i.Furthermore,
(Z1Z2)a = eaLn(z1z2) = ea(Ln z1+Ln z2+2πiN+) = Za
1Za2 e
2πiaN+ , (B.31)(Z1
Z2
)a
= eaLn(z1/z2) = ea(Ln z1−Ln z2+2πiN−) =Za
1
Za2
e2πiaN− , (B.32)
where the integers N± are determined from eq. (A.21).
APPENDIX C: Alternative definitions for Arctan
and Arccot
The well-known reference book for mathematical functions by Abramowitz and Ste-gun (see ref. 1) defines the principal values of the complex arctangent and arccotan-gent functions by employing the principal values of the logarithms in eqs. (1) and (4).This yields,12
Arctan z =1
2iLn
(1 + iz
1− iz
)
, (C.1)
Arccot z = Arctan
(1
z
)
=1
2iLn
(z + i
z − i
)
. (C.2)
Note that with these definitions, the branch cuts are still given by eqs. (14) and (15),respectively. Comparing the above definitions with those of eqs. (12) and (13), onecan check that the two definitions differ only on the branch cuts and at certain pointsof infinity. In particular, there is no longer any ambiguity in how to define Arccot(0).Plugging z = 0 into eq. (C.2) yields
Arccot(0) =1
2iLn(−1) = 1
2π .
12A definition of the principal value of the arccotangent function that is equivalent to eq. (C.2)for all complex numbers z is:
Arccot z =1
2i[Ln(iz − 1)− Ln(iz + 1)] .
A proof of the equivalence of this form and that of eq. (C.2) can be found in Appendix E of the firstreference in ref. 5.
30
It is convenient to define a new variable,
v =i− z
i+ z, =⇒ −1
v=
z + i
z − i. (C.3)
Then, we can write:
Arctan z + Arccot z =1
2i
[
Ln v + Ln
(
−1
v
)]
=1
2i
[
Ln|v|+ Ln
(1
|v|
)
+ iArg v + iArg
(
−1
v
)]
=1
2
[
Arg v +Arg
(
−1
v
)]
. (C.4)
It is straightforward to check that for any non-zero complex number v,
Arg v +Arg
(
−1
v
)
=
{
π , for Im v ≥ 0 ,
−π , for Im v < 0 .(C.5)
Using eq. (C.3), we can evaluate Im v by computing
i− z
i+ z=
(i− z)(−i − z)
(i+ z)(−i + z)=
1− |z|2 + 2iRe z
|z|2 + 1 + 2 Im z.
Writing |z|2 = (Re z)2 + (Im z)2 in the denominator,
i− z
i+ z=
1− |z|2 + 2iRe z
(Re z)2 + (Im z + 1)2.
Hence,
Im v ≡ Im
(i− z
i+ z
)
=2Re z
(Re z)2 + (Im z + 1)2.
We conclude that
Im v ≥ 0 =⇒ Re z ≥ 0 , Im v < 0 =⇒ Re z < 0 .
Therefore, eqs. (C.4) and (C.5) yield:
Arctan z +Arccot z =
12π , for Re z ≥ 0 ,
−12π , for Re z < 0 .
(C.6)
This relation differs from eq. (22) when z lives on one of the branch cuts, i.e. forRe z = 0 and z 6= ±i.
31
One disadvantage of the definition of the principal value of the arctangent givenby eq. (C.1) concerns the value of Arctan(−∞). In particular, if z = x is real,
∣∣∣∣
1 + ix
1− ix
∣∣∣∣= 1 , (C.7)
Since Ln 1 = 0, it would follow from eq. (C.1) that for all real x,
Arctan x = 12Arg
(1 + ix
1− ix
)
. (C.8)
Indeed, eq. (C.8) is correct for all finite real values of x. It also correctly yieldsArctan (∞) = 1
2Arg(−1) = 1
2π, as expected. However, if we take x → −∞ in
eq. (C.8), we would also get Arctan(−∞) = 12Arg(−1) = 1
2π, in contradiction with the
conventional definition of the principal value of the real-valued arctangent function,where Arctan (−∞) = −1
2π. This slight inconsistency is not surprising, since the
principal value of the argument of any complex number z must lie in the range−π < Arg z ≤ π. Consequently, eq. (C.8) implies that −1
2π < Arctan x ≤ 1
2π, which
is inconsistent with eq. (18) as the endpoint at −12π is missing.
Some authors finesse this defect by defining the value of Arctan(−∞) as the limitof Arctan (x) as x → −∞. Note that
limx→−∞
Arg
(1 + ix
1− ix
)
= −π ,
since for any finite real value of x < −1, the complex number (1+ ix)/(1− ix) lies inQuadrant III13 and approaches the negative real axis as x → −∞. Hence, eq. (C.8)yields
limx→−∞
Arctan (x) = −12π .
With this interpretation, eq. (C.1) is a perfectly good definition for the principal valueof the arctangent function.
It is instructive to consider the difference of the two definitions of Arctan z givenby eqs. (12) and (C.1). Using eq. (B.21), it follows that
Ln
(1 + iz
1− iz
)
− [Ln(1 + iz)− Ln(1− iz)] = 2πiN− ,
13This is easily verified. We write:
z ≡ 1 + ix
1− ix=
1 + ix
1− ix· 1 + ix
1 + ix=
1− x2 + 2ix
1 + x2.
Thus, for real values of x < −1, it follows that Rez < 0 and Imz < 0, i.e. the complex number z lies inQuadrant III. Moreover, as x → −∞, we see that Rez → −1 and Imz → 0−, where 0− indicates thatone is approaching 0 from the negative side. Some authors write limx→∞(1+ ix)/(1− ix) = −1− i0to indicate this behavior, and then define Arg(−1− i0) = −π.
32
where
N− =
−1 , if Arg(1 + iz)−Arg(1− iz) > π ,
0 , if −π < Arg(1 + iz)− Arg(1− iz) ≤ π ,
1 , if Arg(1 + iz)±Arg(1− iz) ≤ −π .
(C.9)
To evaluate N− explicitly, we must examine the quantity Arg(1+ iz)−Arg(1− iz) asa function of the complex number z = x+ iy. Hence, we shall focus on the quantityArg(1 − y + ix) − Arg(1 + y − ix) as a function of x and y. If we plot the numbers1 − y + ix and 1 + y − ix in the complex plane, it is evident that for finite values ofx and y and x 6= 0 then −π < Arg(1 − y + ix) − Arg(1 + y − ix) < π . The case ofx = 0 is easily treated separately, and we find that
Arg(1− y)−Arg(1 + y) =
π , if y > 1 ,
0 , if −1 < y < 1 ,
−π , if y < −1 .
Note that we have excluded the points x = 0, y = ±1, which correspond to the branchpoints where the arctangent function diverges.
Therefore, in the finite complex plane excluding the branch points at z = ±i,
N− =
{
1 , if Re z = 0 and Im z < −1 ,
0 , otherwise.
That is, in the finite complex plane, the two possible definitions for the principal valueof the arctangent function given by eqs. (12) and (C.1) differ only on the branch cutalong the negative imaginary axis below z = −i. Indeed, for finite values of z 6= ±i,
1
2iLn
(1 + iz
1− iz
)
=
π +1
2i[Ln(1 + iz) − Ln(1− iz)] , if Re z = 0 and Im z < −1 ,
1
2i[Ln(1 + iz)− Ln(1− iz)] , otherwise .
(C.10)In order to compare the two definitions of Arctan z in the limit of |z| → ∞, we canemploy the relation Arccot z = Arctan(1/z) [which holds for both sets of definitions],and examine the behavior of Arccot z in the limit of z → 0.
The difference of the two definitions of Arccotz given by eqs. (13) and (C.2) followsimmediately from eq. (C.10). For z 6= ±i and z 6= 0,14
1
2iLn
(z + i
z − i
)
=
π +1
2i
[
Ln
(
1 +i
z
)
− Ln
(
1− i
z
)]
, if Re z = 0 and 0 < Im z < 1 ,
1
2i
[
Ln
(
1 +i
z
)
− Ln
(
1− i
z
)]
, otherwise .
(C.11)
14Eq. (C.11) is also valid for |z| → ∞, in which case both definitions of the arccotangent yieldArccot(∞) = 0, independently of the direction in the complex plane in which z approaches complexinfinity. This behavior is equivalent to the statement that both definitions of the principal value ofthe arctangent in eq. (C.10) yield Arctan(0) = 0.
33
We can now derive the behavior of Arccotz when z is a non-zero complex numberinfinitesimally close to z = 0. Using the results of eq. (C.11), it follows that eq. (16)is modified to:
Arccot z =z→0 , z 6=0
{12π , for Re z ≥ 0 ,
−12π , for Re z < 0 .
As expected, Arccot z is discontinuous across the branch cut, which corresponds tothe line in the complex plane corresponding to Re z = 0 and |Im z| < 1. However,Arccot z as defined by eq. (C.2) is a continuous function of z along the branch cut,with
limy→0+
Arccot(iy) = limy→0−
Arccot(iy) = 12π . (C.12)
This is in contrast to the behavior of Arccot as defined in Mathematica 8 [cf. eq. (13)],where the value of Arccot is discontinuous at z = 0 on the branch cut, in which caseone must separately define the value of Arccot(0).
So which set of conventions is best? Of course, there is no one right or wronganswer to this question. The authors of refs. 4–6 argue for choosing eq. (12) to definethe principal value of the arctangent and eq. (C.2) to define the principal value ofthe arccotangent. This has the benefit of ensuring that eq. (C.12) is satisfied so thatArccot(0) is unambiguously defined. But, it will lead to corrections to the relationArccotz = Arctan(1/z) for a certain range of complex numbers that lie on the branchcuts. In particular, with the definitions of Arctan z and Arccot z given by eqs. (12)and (C.12), we immediately find from eq. (C.11) that
Arccot z =
π +Arctan
(1
z
)
, if Re z = 0 and 0 < Im z < 1 ,
Arctan
(1
z
)
, otherwise ,
excluding the branch points z = ±i where Arctan z and Arccot z both diverge.Likewise, with the definitions of Arctan z and Arccot z given by eqs. (12) and (C.12),the expression for Arctan z + Arccot z [given in eqs. (22) and (C.6)] will also bemodified on the branch cuts,
Arctan z +Arccot z =
12π , for Re z > 0 ,
12π , for Re z = 0 , and Im z > −1 ,
−12π , for Re z < 0 ,
−12π , for Re z = 0 , and Im z < −1 .
(C.13)
A similar set of issues arise in the definitions of the principal values of the in-verse hyperbolic tangent and cotangent functions. It is most convenient to definethese functions in terms of the corresponding principal values of the arctangent andarccotangent functions following eqs. (53) and (54),
Arctanh z = iArctan(−iz) , (C.14)
Arccoth z = iArccot(−iz) . (C.15)
34
As a practical matter, I usually employ the Mathematica 8 definitions, as this is aprogram that I use most often in my research.
� CAUTION!!
The principal value of the arccotangent is given in terms the principal value ofthe arctangent,
Arccot z = Arctan
(1
z
)
, (C.16)
for both the Mathematica 8 definition [eq. (13)] or the alternative definition presentedin eq. (C.2). However, many books define the principal value of the arccotangentdifferently via the relation,
Arccot z = 12π −Arctan z . (C.17)
This relation should be compared with the corresponding relations, eqs. (22) and(C.6), which are satisfied with the definitions of the principal value of the arccotangentintroduced in eqs. (13) and (C.2), respectively. Eq. (C.17) is adopted by the Maple16 computer algebra system, which is one of the main competitors to Mathematica.
The main motivation for eq. (C.17) is that the principal interval for real values x isgiven by 0 ≤ Arccotx ≤ π , instead of the interval quoted in eq. (20). One advantageof this latter definition is that the real-valued arccotangent function, Arccot x, iscontinuous at x = 0, in contrast to eq. (C.16) which exhibits a discontinuity at x = 0.Note that if one adopts eq. (C.17) as the the definition of the principal value of thearccotangent, then the branch cuts of Arccot z are the same as those of Arctan z,namely eq. (14). The disadvantages of the definition given in eq. (C.17) are discussedin refs. 4 and 5.
Which convention does your calculator and/or your favorite mathematics softwareuse? Try evaluating Arccot(−1). In the convention of eq. (13) or eq. (C.2), we haveArccot(−1) = −1
4π, while in the convention of eq. (C.17), we have Arccot(−1) = 3
4π.
APPENDIX D: Derivation of eq. (22)
To derive eq. (22), we will make use of the computations provided in Appendix C.Start from eq. (C.6), which is based on the definitions of the principal values of thearctangent and arccotangent given in eqs. (C.1) and (C.2), respectively. We then useeqs. (C.10) and (C.11) which allow us to translate between the definitions of eqs. (C.1)and (C.2) and the Mathematica 8 definitions of the principal values of the arctangentand arccotangent given in eqs. (12) and (13), respectively. Eqs. (C.10) and (C.11)imply that the result for Arctan z + Arccot z does not change if Re z 6= 0. For thecase of Re z = 0, Arctan z + Arccot z changes from 1
2π to −1
2π if 0 < Im z < 1 or
Im z < −1. This is precisely what is exhibited in eq. (22).
35
APPENDIX E: Proof that Re (±iz +√1 − z2) > 0
It is convenient to define:
v = iz +√1− z2 ,
1
v=
1
iz +√1− z2
= −iz +√1− z2 .
In this Appendix, we shall prove that:
Re v ≥ 0 , and Re
(1
v
)
≥ 0 . (E.1)
Using the fact that Re (±iz) = ∓Im z for any complex number z,
Re v = −Im z + |1− z2|1/2 cos[12Arg(1− z2)
], (E.2)
Re
(1
v
)
= Im z + |1− z2|1/2 cos[12Arg(1− z2)
]. (E.3)
One can now prove that
Re v ≥ 0 , and Re
(1
v
)
≥ 0 , (E.4)
for any finite complex number z by considering separately the cases of Im z < 0,Im z = 0 and Im z > 0. The case of Im z = 0 is the simplest, since in this caseRe v = 0 for |z| ≥ 1 and Re v > 0 for |z| < 1 (since the principal value of thesquare root of a positive number is always positive). In the case of Im z 6= 0, wefirst note that that −π < Arg(1− z2) ≤ π implies that cos
[12Arg(1− z2)
]≥ 0. Thus
if Im z < 0, then it immediately follows from eq. (E.2) that Re v > 0. Likewise, ifImz > 0, then it immediately follows from eq. (E.3) that Re (1/v) > 0. However, thesign of the real part of any complex number z is the same as the sign of the real partof 1/z, since
1
x+ iy=
x− iy
x2 + y2.
Hence, it follows that both Re v ≥ 0 and Re (1/v) ≥ 0, and eq. (E.1) is proven.
APPENDIX F: Derivation of eq. (68)
We begin with the definitions given in eqs. (39) and (66),15
iArccos z = Ln(
z + i√1− z2
)
, (F.1)
Arccosh z = Ln(
z +√z + 1
√z − 1
)
, (F.2)
15We caution the reader that some authors employ different choices for the definitions of theprincipal values of arccos z and arccosh z and their branch cuts. The most common alternativedefinitions are:
Arccosh z = iArccos z = Ln(z +√
z2 − 1) ,
36
where the principal values of the square root functions are employed following thenotation of eq. (32). Our first task is to relate
√z + 1
√z − 1 to
√z2 − 1. Of course,
these two quantities are equal for all real numbers z ≥ 1. But, as these quantities areprincipal values of the square roots of complex numbers, one must be more careful inthe general case. Employing eq. (B.32) to the principal value of the complex squareroot function yields:16
√z1z2 = e
12Ln(z1z2) = e
12 (Ln z1+Ln z2+2πiN+) =
√z1√z2 e
πiN+ ,
where
N+ =
−1 , if Arg z1 +Arg z2 > π ,
0 , if −π < Arg z1 +Arg z2 ≤ π ,
1 , if Arg z1 +Arg z2 ≤ −π .
That is, √z1z2 = ε
√z1√z2 , ε = ±1 , (F.3)
where the choice of sign is determined by:
ε =
{
+1 , if −π < Arg z1 +Arg z2 ≤ π ,
−1 , otherwise.
Thus, we must determine in which interval the quantity Arg(z + 1) + Arg(z − 1)lies as a function of z. The special cases of z = ±1 must be treated separately, sinceArg 0 is not defined. By plotting the complex points z + 1 and z − 1 in the complexplane, one can easily show that for z 6= ±1,
−π < Arg (z + 1) + Arg (z − 1) ≤ π , if
Im z > 0 and Re z ≥ 0 ,
or
Im z = 0 and Re z > −1 ,
or
Im z < 0 and Re z > 0 .
If the above conditions do not hold, then Arg(z + 1) + Arg(z − 1) lies outside therange of the principal value of the argument function. Hence, we conclude that ifz1 = z + 1 and z2 = z − 1 then if Im z 6= 0 then ε in eq. (F.3) is given by:
ε =
{
+1 , if Re z > 0 , Im z 6= 0 or Re z = 0 , Im z > 0 ,
−1 , if Re z < 0 , Im z 6= 0 or Re z = 0 , Im z < 0 .
which differ from the definitions, eqs. (F.1) and (F.2), employed by Mathematica 8 and these notes.In particular, with the alternative definitions given above, Arccos z now possesses the same set ofbranch cuts as Arccosh z given by eq. (70), in contrast to eq. (42). Moreover, Arccos z no longersatisfies eq. (38) if either (Re z)(Im z) < 0 or if |Re z| > 1 and Im z = 0 [cf. eq. (F.6)]. Otherdisadvantages of the alternative definitions of Arccos z and Arccosh z are discussed in ref. 4.
16Following the convention established above eq. (32), we employ the ordinary square root sign(√
) to designate the principal value of the complex square root function.
37
In the case of Im z = 0, we must exclude the points z = ±1, in which case we alsohave
ε =
{
+1 , if Im z = 0 and Re z > −1 with Re z 6= 1 ,
−1 , if Im z = 0 and Re z < −1 .
It follow that Arccosh z = Ln(z±√z2 − 1), where the sign is identified with ε above.
Noting that z −√z2 − 1 = [z +
√z2 − 1]−1, where z +
√z2 − 1 is real and negative
if and only if Im z = 0 and Re z ≤ −1,17 one finds after applying eq. (36) that:
Ln(z −√z2 − 1) =
{
2πi− Ln(z +√z2 − 1) , for Im z = 0 and Re z ≤ −1 ,
−Ln(z +√z2 − 1) , otherwise .
To complete this part of the analysis, we must consider separately the points z = ±1.At these two points, eq. (F.2) yields Arccosh(1) = 0 and Arccosh(−1) = Ln(−1) = πi.Collecting all of the above results then yields:
Arccoshz =
Ln(z +√z2 − 1) , if Im z > 0 , Re z ≥ 0 or Im z = 0 , Re z ≥ −1
or Im z < 0,Re z > 0 ,
−Ln(z +√z2 − 1) , if Im z > 0 , Re z < 0 or Im z < 0 , Re z ≤ 0 ,
2πi− Ln(z +√z2 − 1) , if Im z = 0,Re z ≤ −1 .
(F.4)Note that the cases of z = ±1 are each covered twice in eq. (F.4) but in both respectivecases the two results are consistent.
Our second task is to relate i√1− z2 to
√z2 − 1. To accomplish this, we first
note that for any non-zero complex number z, the principal value of the argument of−z is given by:
Arg(−z) =
{
Arg z − π , if Arg z > 0 ,
Arg z + π , if Arg z ≤ 0 .(F.5)
This result is easily checked by considering the locations of the complex numbers zand −z in the complex plane. Hence, by making use of eqs. (32) and (F.5) along withi = eiπ/2, it follows that:
i√1− z2 =
√
|z2 − 1|e 12 [π+Arg(1−z2)] = η
√z2 − 1 , η = ±1 ,
where the sign η is determined by:
η =
{
+1 , if Arg(1− z2) ≤ 0 ,
−1 , if Arg(1− z2) > 0 ,
17Let w = z+√z2 − 1, and assume that Im w = 0 and Re w 6= 0. That is, w is real and nonzero,
in which case Im w2 = 0. But
0 = Im w2 = Im[
2z2 − 1 + 2z√
z2 − 1]
= Im (2zw − 1) = 2wIm z ,
which confirms that Im z = 0, i.e. z must be real. If we require in addition that that Re w < 0,then we also must have Re z ≤ −1.
38
assuming that z 6= ±1. If we put z = x + iy, then 1 − z2 = 1 − x2 + y2 − 2ixy, andwe deduce that
Arg(1− z2) is
positive , either if xy < 0 or if y = 0 and |x| > 1 ,
zero , either if x = 0 or if y = 0 and |x| < 1 ,
negative , if xy > 0 .
We exclude the points z = ±1 (corresponding to y = 0 and x = ±1) where Arg(1−z2)is undefined. Treating these two points separately, eq. (F.1) yields Arccos(1) = 0 andiArccos(−1) = Ln(−1) = πi. Collecting all of the above results then yields:
iArccosz =
Ln(z +√z2 − 1) , if Im z > 0 , Re z ≥ 0 or Im z < 0 , Re z ≤ 0
or Im z = 0 , |Re z| ≤ 1 ,
−Ln(z +√z2 − 1) , if Im z > 0 , Re z < 0 or Im z < 0 , Re z > 0 ,
or Im z = 0 , Re z ≥ 1 ,
2πi− Ln(z +√z2 − 1) , if Im z = 0 , Re z ≤ −1 .
(F.6)Note that the cases of z = ±1 are each covered twice in eq. (F.6) but in both respectivecases the two results are consistent.
Comparing eqs. (F.4) and (F.6) established eq. (68) and our proof is complete.
References
1. A comprehensive treatment of the properties of the inverse trigonometric andinverse hyperbolic functions can be found in Milton Abramowitz and Irene A. Stegun,Handbook of Mathematical Functions (Dover Publications, Inc., New York, 1972).
2. Michael Trott, The Mathematica Guidebook for Programming (Springer Sci-ence+Business Media, Inc., New York, NY, 2004). In particular, see section 2.2.5.
3. A.I. Markushevich, Theory of Functions of a Complex Variable, Part I (AMSChelsea Publishing, Providence, RI, 2005)
4. W. Kahan, “Branch Cuts for Complex Elementary Functions,” in The State of
Art in Numerical Analysis, edited by A. Iserles and M.J.D. Powell (Clarendon Press,Oxford, UK, 1987) pp. 165–211.
5. R.M. Corless, D.J. Jeffrey, S.M. Watt and J.H. Davenport, “According toAbramowitz and Stegun or arccoth needn’t be uncouth,” ACM SIGSAM Bulletin34, 58–65 (2000); J.H. Davenport, “According to Abramowitz & Stegun (II),” 2006[available from http://staff.bath.ac.uk/masjhd/2Nov-2.pdf].
6. R. Bradford, R.M. Corless, J.H. Davenport, D.J. Jeffrey and Stephen M. Watt,“Reasoning about the elementary functions of complex analysis,” Annals of Mathe-matics and Artificial Intelligence 36, 303–318 (2002).
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